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Sasaki-Einstein orbits in compact Hermitian symmetric spaces

Yuuki Sasaki

TL;DR

The paper characterizes isotropy orbits of irreducible Hermitian symmetric spaces of compact type as CR submanifolds, detailing the decomposition into complex and totally real distributions and proving leaves of the totally real foliation are totally geodesic in the orbit. It provides a complete classification of contact and Sasaki structures on these orbits, identifies when an orbit is Sasaki (and when Sasaki–Einstein) via restricted root data, and determines the integrability of the complex distribution. Across the classical families, it derives explicit conditions on the ambient metric (through the invariant parameter $d$) that yield Sasaki–Einstein orbits and lists when such orbits exist, including cases where none exist (e.g., CP$^{n-1}$ and certain Grassmannians). The work markedly connects CR/submanifold geometry with the Lie-theoretic structure of the restricted root system, yielding a detailed map from root data to geometric properties of isotropy orbits and their leaves. These results illuminate the geometry of isotropy actions on compact Hermitian symmetric spaces and provide concrete invariants for identifying special CR, Sasaki, and Sasaki–Einstein orbits.

Abstract

The aim of the present papar is to study the orbits of the isotropy gourp action on an irreducible Hermitian symmetric space of compact type. Specifically, we examine the properties of these orbits as {\it CR} submanifolds of a Kähler manifold. Our focus is on the leaves of the totally real distribution, and we investigate the properties of leaves as a Riemannian submanifold. In particular, we prove that any leaf is a totally geodesic submanifold of the orbit. Additionally, we explore the conditions under which each leaf becomes a totally geodesic submanifold of the ambient space. The integrability of the complex distribution is also studied. Moreover, we analyze a contact structure of orbits where the rank of the totally real distribution is 1. We obtain a classification of the orbits that possess either a contact structure or a Sasakian structure compatible with the complex structure on the ambient space. Furthermore, we classify those Sasaki orbits that are Einstein with respect to the induced metric. Specifically, we completely detemine Sasaki-Einstein orbits.

Sasaki-Einstein orbits in compact Hermitian symmetric spaces

TL;DR

The paper characterizes isotropy orbits of irreducible Hermitian symmetric spaces of compact type as CR submanifolds, detailing the decomposition into complex and totally real distributions and proving leaves of the totally real foliation are totally geodesic in the orbit. It provides a complete classification of contact and Sasaki structures on these orbits, identifies when an orbit is Sasaki (and when Sasaki–Einstein) via restricted root data, and determines the integrability of the complex distribution. Across the classical families, it derives explicit conditions on the ambient metric (through the invariant parameter ) that yield Sasaki–Einstein orbits and lists when such orbits exist, including cases where none exist (e.g., CP and certain Grassmannians). The work markedly connects CR/submanifold geometry with the Lie-theoretic structure of the restricted root system, yielding a detailed map from root data to geometric properties of isotropy orbits and their leaves. These results illuminate the geometry of isotropy actions on compact Hermitian symmetric spaces and provide concrete invariants for identifying special CR, Sasaki, and Sasaki–Einstein orbits.

Abstract

The aim of the present papar is to study the orbits of the isotropy gourp action on an irreducible Hermitian symmetric space of compact type. Specifically, we examine the properties of these orbits as {\it CR} submanifolds of a Kähler manifold. Our focus is on the leaves of the totally real distribution, and we investigate the properties of leaves as a Riemannian submanifold. In particular, we prove that any leaf is a totally geodesic submanifold of the orbit. Additionally, we explore the conditions under which each leaf becomes a totally geodesic submanifold of the ambient space. The integrability of the complex distribution is also studied. Moreover, we analyze a contact structure of orbits where the rank of the totally real distribution is 1. We obtain a classification of the orbits that possess either a contact structure or a Sasakian structure compatible with the complex structure on the ambient space. Furthermore, we classify those Sasaki orbits that are Einstein with respect to the induced metric. Specifically, we completely detemine Sasaki-Einstein orbits.

Paper Structure

This paper contains 16 sections, 60 theorems, 192 equations, 2 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. $CR$ submanifolds
  4. Compact Hermitian symmetric spaces
  5. The restricted root system
  6. The complex structure and the restricted root system
  7. Isotropy orbits as a $CR$ submanifold
  8. Case. 1
  9. Case. 2
  10. Contact $CR$ orbits
  11. Contact $CR$ orbits
  12. Some preliminaries I
  13. Sasaki $CR$ orbits
  14. Sasaki-Einstein $CR$ orbits
  15. Some preliminaries II
  16. ...and 1 more sections

Key Result

Lemma 2.2

Chen For any $CR$ submanifold, the totally real distribution $\mathcal{D}^{\perp}$ is integrable. Moreover, the complex distribution $\mathcal{D}$ is integrable if and only if for any $x \in N, X,Y \in \mathcal{D}, Z \in \mathcal{D}^{\perp}$.

Theorems & Definitions (99)

  • Definition 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • Lemma 2.7
  • Lemma 2.8
  • Lemma 2.9: The Gauss equation
  • Lemma 2.10
  • ...and 89 more